Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve

Let $E:y^2=x^3-x$. We prove that $\mathrm{\#}E({\mathbf{F}}_p)\mathrm{/}8$ is squarefree and has at most two prime factors for $\gg X\mathrm{/}(\log X{)}^2$ primes $p\le X$. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal $\mathfrak{a}$, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri–Vinogradov estimate to rational norm divisibility at level $1\mathrm{/}2$, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen’s weighted sieve. The primes counted may be restricted to $p\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$; their normalized orders are odd, squarefree, and supported on primes congruent to $1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.